Pacific Journal of

THE RANGE OF A NORMAL DERIVATION

BARRY E.JOHNSONAND JAMES PATRICK WILLIAMS

Vol. 58, No. 1 March 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 58, No. 1, 1975

THE RANGE OF A NORMAL DERIVATION

B. E. JOHNSON AND J. P. WILLIAMS

The inner derivation δA implemented by an element A of the algebra MVM) of bounded linear operators on the separable Hubert space W is the map X -* AX - XA (X G ® (W)). The main result of this paper is that when A is normal, range ι inclusion Jί(δB) C M(δA) is equivalent to the condition that B=f(A) where Λ(z, w) = (/(z)-/(w))(z - w)"1 (taken as 0 when z = w) has the property that Λ(z, w)t(z, w) is a class kernel on L2(μ) whenever ί(z, w) is such a kernel. Here μ is the dominating scalar valued spectral measure of A constructed in multiplicity theory. In order that a Borel function / satisfy this condition it is necessary that / be equal almost everywhere to a Lipschitz function with derivative in σ(A) at each limit point of σ(A) and it is sufficient (for A self-adjoint) that / G C(3)(R).

For such operators B there is a>o a factorization δB = δAτ = τδA by an ultraweakly continuous r from 38($?) into itself satisfying

τ(A']XA'2) = A\τ(X)A'2 for Xe»(I) and A\,A'2 commuting with A. When 2C is finite dimensional (X, Y) = trace (XY*) is an inner product and 38(20 = 9t(δA)®{A*}' is the orthogonal direct sum of the range of δA and {A *}' = { Y G 38 (30 : YA * = A * Y}, the commutant of the adjoint of A. This simple fact suggests that ^(δ^) is a natural subspace, like the commutant, associated with A. The orthogonal decomposition also shows that range inclusion &ί{δB) C0ί(δA) holds for a pair of operators if and only if B G {A}'\ or equivalently, if and only if

B is a polynomial in A. In this case δB = δAτ = τδA with r as above. When $f is infinite dimensional the situation is less clear. We do not know whether 9t (δA) Π {A *}' = 0 in general. The sum

S2(SA) + {.4*}' is always weakly dense in 38 (X) but is rarely closed; in fact for A normal it is closed if and only if A has a finite specturm [1].

The condition BG{A}" is neither sufficient for 0l{δB)C0i{δA) (even if A is positive and compact [19]), nor necessary [Yang Ho, private communication]. If A G 38 (20 and B = /(A-), where / is analytic in a neighborhood of the specturm of A, then 3#(δB) C<3i(δA) but range inclusion does not imply B = f(A) for some analytic / [20]. Finally, if {A}' contains no nonzero operator then the norm closure of 3l{δA) contains the ideal of compact operators [22]. In this case there are operators

B£ {A}" with 0l(δB)CSk(δA)". There are normal operators A with this

105 106 B. E. JOHNSON AND J. P. WILLIAMS property (multiplication by x in L2(0, 1) for example) and compact operators [16] but none that is both normal and compact. (See the remark following (2.1) below). As byproducts of our study of the range of a normal derivation we obtain improvements of the results of [19, 20] and a simpler proof of the theorem of [1] mentioned above. Our results also yield solutions to some asymptotic commutativity problems raised in [3].

1. Auxiliary results. If E,F,G are Banach spaces, SG 35 (F, G) and T E 35(F, G), then the implies that Θi(S) C$(Γ) if and only if there is JR E 33(F,F/Ker(Γ)) with S = TR, where f is the element of 3S(F/Ker(Γ),G) associated with T. In < particular, range inclusion 3l{δB)C@l(δA) of derivations on 33(20 amounts to a factorization 8B = 8Aσ with σ a bounded linear operator from 3δ($0 into 3δ($0/{Λ}'. Our first goal is to show that if A is normal this trivial factorization can be sharpened to: δβ = 8Aσ for some ultraweakly continuous linear operator σ from 39 (ffl) into itself. For this and later applications we need some simple facts about range inclusion in general.

LEMMA (1.1). // S,T E 3δ(F, F) the following are equivalent: (1) There exists a constant c such that \\Sx ||g c \\Tx \\ for all xGE. (2) There exists a constant c such that for each fEF* there is a gEF* with \\g || ^ c 11/11 and S*f = T*g. (3) $(S*)$(Γ*)

Proof Suppose that (1) holds, and fix / E F*. Then Tx -+ (5x, /) extends to a bounded linear functional on F by the Hahn-Banach theorem and therefore there is a vector g E F* with ||g || S c \\f\\ such that (SxJ) = (Tx,g) for all x E E. Hence 5*/= Γ*g so that (2) is satisfied. Clearly (2) implies (3). Suppose that (3) holds. If fEF* then

S*/=Γ*g f0Γ some g<ΞF* and therefore |(Sx,/)| = || ^ c ||7JC ||||/|| for all x E E and all /E F*. Then ||5x \\^c\\Tx \\ so that (1) holds.

COROLLARY (1.2). J/S, ΓE S8(F,F) then0l{S**)cgi{T**) if and only ifSk{S)C0i{T^) where<31{S) is identified with its canonical image in F**.

Proof. Since S** is an extension of S the necessity is trivial. Suppose 3?(S) C^(Γ**). If JC E F then there is ξ E E** with THE RANGE OF A NORMAL DERIVATION 107

Sx = T**ξ and hence |l = |<Γ *£J>|S||£||||Γ /II for fEF*. Hence by the uniform boundedness theorem there is a con- stant c such that |

COROLLARY (1.3). 7/ S, Γ E 3&(E9 F) these are equivalent: (1) ^(S*)$(Γ*) (2) (3)

Proof. Conditions (2) and (3) are equivalent by Corollary (1.2). Also (1) trivially implies (2). Suppose (2) holds. If / E F* then

S*f = j***ξ for some ξ^F***. Now each such ψ has the form

£ = £o + £i where ξ0 E F° and ξ, F*. Also Γ*** extends T* and maps

F°into F° and so 5*/- Γ*f, = Γ***£0E F* Π F° = {0}. Thus 5*/ = Γ*£i E 3?(Γ*). Therefore (2) implies (1). In the next result and in several subsequent arguments we shall make use of the duality relations between the % = JK{^€) of compact operators on Sίf, equipped with the usual sup norm, and the Banach space 3~ = S'i'X) of trace class operators on $?, equipped with the trace norm || ||^. Recall [4, 14] that ?f may be isometrically identified with the conjugate space of X and that 33 ($ί) is the conjugate space of 3~. The canonical here is (X, Γ) = trace (XT) = trace (TX) for T E SΓ and X belonging to either 3δ (?O or to X. Finally, the on 38 (X) is the weak* topology σ(

COROLLARY (1.4). These are equivalent for A,B E

(1) There exists a constant c such that \\ δB (X) || g c || δΛ (X) || for all

(2) There exists a constant c such that for each T E SΓ{%) there is

SΓ(W) with \\Sy^c\\T\\rindδB(T) = (3) (4)

Proof Conditions (3) and (4) are equivalent by Corollary (1.3) with S = (δa IX) and T = (δA\ X). Also (1) and (3) and (2) and (4) are equivalent by Lemma (1.1).

COROLLARY (1.5). These are equivalent for A,B E (1) (2) (3) There exists a bounded linear map σ from $($?) into 108 B. E. JOHNSON AND J. P. WILLIAMS

St(X)l{A}' such that δB = δAσ where δA : &(%)l{A}'->9t(δA) is the canonical map associated with δA.

(4) There exists a constant c such that ||δβ(Γ)||j ^ c \\δA(T)\\

Proof. Conditions (1) and (4) are equivalent by Lemma (1.1) applied to S = (δB\2Γ) and T = (δA\SΓ). Conditions (1) and (2) are equivalent by Corollary (1.2) applied to the restrictions of δA and δB to 3ΐ, and (1) and (3) are equivalent by the remark preceding Lemma (1.1).

2. Normal derivations. In this section we show that if A is a normal operator on a Sίf (assumed to be separable here and in the remainder of the paper) then range inclusion 3ί{δB) C9l(δA) holds only for operators B E{A}". We use the fact that there is a projection P of norm one from 33(20 onto {A}' with the property

P(A5X4 J) = A \P{X)A'2 for X G &(%) and A \9A'2 in {A}'. In fact any projection of norm one onto the commutant has this commutativity property [17]. The existence of such projections is a standard fact in the theory of von Neumann algebras [2; Chapter 2]. A simple way to obtain one is to choose a V with {V}' = {A}' and set P(X) = glim^oc V*nXVn where glim is a fixed generalized limit on /" and the equality is in the weak (inner product) sense. (See [23])

The commutativity property of P immediately implies 0l(δA)C $fc(l - P) but one does not have equality here in general since for A normal, $l(δA) + {A}' is norm closed only in the trivial case in which the spectrum of A is finite [1]. The following fact is sufficient for our needs here:

LEMMA (2.1). ί%(δΛ) and ^(1 - P) have same ultraweak closures. Hence if Aξ = λξ and Aη = λη for ξ, η E X then ((1 - P)(X)ξ, η) = 0.

Proof. We have 0l{δA)C^ί{\-P) so that by considering an- nihilators in SΓ(^) it suffices to show that ((1 - P)(X), Γ> = 0 for each trace class operator T that commutes with A. Now for such a T we have the polar factorization T = ί7(Γ*Γ)^ where U is a partial isometry and both factors belong to {A}'. Since P(XU) = P(X)U it suffices to consider the case T ^ 0. In fact, by the we need only show that <(1 - P)(X),E) = 0 for X G 98(30 and E E{A}1 a projection of finite rank. Now for the projection P constructed in [12] this can easily be verified since P(X) is obtained from the operators V*XV with V unitarv in {A}". However we can also prove the assertion assuming only th< existence of P as follows: With respect to the decomposition ^ = 3$ ':)θ^(£)i=%θ^ι simple calculations with two by two operatr matrices show that P induces a norm one projection p from THE RANGE OF A NORMAL DERIVATION 109

'Q onto the commutant of A0 = (A\ 3€0) such that (X - P(X), E) = trace(Xo-p(Xo)), where X0 = E(X\W0), and this last quantity is 0 f because the formula &(%*) = 9%(δΛo) + {A$} shows that 9t{\-p) =

$(δAo), that is, Xo-p(Xo) is a commutator. The second assertion of the Lemma follows from the first by obserivng that the operator ξ (g) η defined by (ξg) η)(ζ) = (ζ9 η)ξ is a trace class operator commuting with A so that 0 =

REMARK. A similar duality argument shows that if A is normal and compact and if B G 38(30, then »(δB)C»(δΛ)" if and only if B -λ G (A}" Π 3ίf for some scalar A. The next result appears in [7; Theorem 3.2]:

LEMMA (2.2). If A is a normal operator on 3€ then

n » {o} zee

Although we shall make no use of the fact, it is worth observing that (2.2) implies a stronger version of itself.

COROLLARY (2.3). Let μ be a (positive, regular) Borel measure on C with compact support and let A be the operator f(z)-+zf(z) on L\μ). If fE0l(A-zl) for μ almost every z EC then / = 0.

Proof. Let {Kn} be a sequence of compact sets with limμ(KJ =

|| μ || and / G 98 (Λ - zl) for all z G Kn. If Pn is the spectral projection corresponding to Kn then PjE^iiPJiA - λI)Pn) for all λEKn and also for λ EC\Kn because σ(PnA\PnH)CKn. Thus Pnf=0 for all n and consequently / = 0. We shall also need the following result of Korotkov. For a proof see [10, 21]:

LEMMA (2.4). Let μ be a Borel measure on C of compact support If ΓGS3(L2(μ)) has &(T)CL"(μ) then T is a Hilbert- Schmidt operator with kernel t G L\μ x μ) satisfying

ess sup i |ί(z, H>)Nμ(H>)=iK2

where K is the norm of T as an operator from V to L00.

We come now to the main result of this section. ΠO B. E. JOHNSON AND J. P. WILLIAMS

THEOREM (2.5). // A is a normal operator on X then contains no nonzero left or right ideal of 3&(W). Hence if B E and m(δB)C0l(8A) then B <Ξ{A}".

Proof Observe first that for any S, Te®(%) the identity

Xδs(T') = δs(XT') - δs(X)T implies that if &(δs) C0l(δτ) then 9t(δτ) must contain the left (and dually, also the right) ideal of 58(20 generated r by δs(T ) for each T" commuting with T. Hence if ί%(δτ) is known not to contain any left or right ideal ideals then 5 E {Γ}". Thus the second assertion of the theorem is a simple consequence of the first. For ξ, η E $? let ξ g) η denote the operator ζ -H> (ζ, η )ξ. Every left ideal contains an ideal $f(g)τj and so it is enough to show that

Sίf (g)τj C*3l(δA) implies η = 0. (The assertion for right ideals follows on taking adjoints.) By restricting to the smallest reducing subspace of A that contains η we can suppose that A is the operator f(z)-+zf(z) in ft = L2(μ) for some regular Borel measure μ on C of compact support. Let P be a projection of norm 1 from 38($?) onto {A}'. For ξEL2(μ) let γ(ξ) be the unique element of &(%) with

The operator γ is continuous by the closed graph theorem and if

Mh E

2 If ft eL», £EL (μ) then \\Mhy(ψ\\ = ||γ(Λ)ίNllrll IIΛ ||2||^ ||2 so that /ι -^Mhγ(l)£ = ft γ(l)^ is continuous in the V norm, and therefore y (l)ξ E L°°(μ) with ||γ(l)f ||β ^ || γ || ||ξ ||2. By Lemma (2.4) the operator γ(l) is Hilbert-Schmidt with kernel t(z,w) satisfying ess sup/|ί(z, w)\2dμ(w) = X2

for z ELE. Since δA(y(l))ξ = (1 ® η)ξ = (£, 17), the Cauchy-Schwarz inequality gives / r \ / r \

for zEE. Therefore \(ξ9η)\^K\\(A-zI)ξ\\ = K\\(A-zl)*ξ\\almost everywhere, and consequently, for all z Eσ(A) by continuity of the THE RANGE OF A NORMAL DERIVATION 111 right side of this inequality. It follows from this that η E0l(A - zl) for each z G C (see the proof of Lemma (1.1)) and therefore η = 0 by Lemma (2.2).

COROLLARY (2.6). Let A be a normal operator on X. If B G

35($f) and 0l(δB)Cίft(δA) then δB = δAσ = σδA for some ultraweakly continuous linear operator σ from 58($?) into itself with the property f σ(A\XA'2) = A\σ{X)A 2 forXG^(X) and A\, A 2 commuting with A.

Proof Suppose 3#(δβ) C3#(δΛ). Then by (1.5) we can factor

δB = δAτ0 for some r0: &(%)-+&(2e)l{A}'. Making use of a projec- tion P of norm 1 from £$ (SO onto {A}' we can replace τ0 by τ E

&(&(%)) to get δB = δAτ. Thus for XG 38(20, τ(X) is the unique operator satisying P(r(X)) = 0, δΛ(τ(X)) = δβ(X). Since B G {A}" it is easy to check that r inherits the commutativity properties of P, that is

We now replace T by an ultraweakly continuous σ with the desired properties by the following device: the map (τ\X) from JC into $($0 = 3ίf** has adjoint (τ\X)* from 3ίf*** into X*. For ΓGJ = %* C3ίf*** we therefore obtain an operator a{T) = (r |35f)*(Γ) in ST. It is clear that a G 38(5"), that (δA \ 3~)a =a(δA\SΓ) = (δB | J") and that

δβ = α*δΛ = δΛα:*. Since α* G S8(Sδ(Sf)) agrees with r on 3ίT we also have a*(A\XA'2) = A\a*(X)A2, first for X G 5ίf, then by ultraweak continuity of α* and of multiplication by a fixed element of έS($?), for all X in the ultraweak closure of 5ίΓ which is 38(5ίf). Thus σ = α* satisfies the requirements of the Corollary.

3. On a an operator B in the second commutant of a normal operator A must be a bounded Borel function of A. In this section we determine which such functions are admissible for range inclusion of the corresponding derivations. For this we need to develop some background information about Hadamard multipliers.

DEFINITION (3.1). A matrix (γ0) is a Hadamard multiplier of if there is an orthonormal basis {£} of 3€ such that for each X G there is an operator Γ(X) G 38 (30 with (Γ(X)& £) = γίy (X& £) for all /, j.

Thus (γ/;) is a //-multiplier if the Hadamard product of (γt7) and the matrix of any is again the matrix of a bounded operator. I. Schur [15] gives several sufficient conditions that (γί7) be an //-multiplier, among them being that (γ0) is itself the matrix of an operator.

LEMMA (3.2). (1) The condition that (γl7) be an H-multiplier is independent of the choice of orthonormal basis of $f. 112 B. E. JOHNSON AND J. P. WILLIAMS

(2) // (γ0) is an H-multiplier then X—»Γ(X) defines a bounded linear operator on 35 ($?). Moreover Γ is ultraweakly continuous and is the adjoint of the H-multiplier on ίf($f) induced by the transpose of (γ,7).

(3) Let {Γn} be a sequence of H-multipliers. If supjΓj < » and if (Γn(X)ξh fi)-> α<,(Λ£, ξ,) for each Xe®(W) and each i,j then (α/y) is an H-multiplier.

a = (4) Let a \Q γ >' Then (ai}) is not an H-multiplier on for π = /2(Z) or for W = /2(Z+).

Proof (1) If {TJ,} is another orthonormal basis of $? and if

(Γ(X)ηh η,) = γ^Xηy,i,,) then Γ(X) = U*Γ(UXU*)U where ί/ is the unitary operator defined by Lfy; = ξh (2) That Γ is bounded on 35 ($0 is a simple consequence of the closed graph theorem. The other two assertions of (2) are easy to verify. (3) The hypotheses imply that for each XE$(3€) the map

T -*limn(Γn(X)9T) is a bounded linear functional on the subspace of ίT(ffl) consisting of finite linear combinations of the operators ξi

(4) Consider first the case in which (αί7) is a doubly infinite matrix iθ iiθ 2 (ί,/GZ) and let ξ}(e ) = e be the usual basis of L (0,2τr). If Mφ denotes the operator f-*φ f on V for a given φ EL00, then the

Hadamard product of (αiy) and the matrix of Mψ is the matrix associated with MΦ where φ is the function whose Fourier coefficients (φ,ξn) vanish for n < 0 and agree with those of

Consider now the matrix βi} = 1 or 0 depending on whether or not i^ j for i,jGZ+. Then (&) cannot be an //-multiplier of /2(Z+) because the doubly infinite matrix (α(/) just mentioned is the sum of the direct sum (β^φίβ-,,,) and a matrix which is an obvious //-multiplier of /2(Z). There is another, perhaps more natural, way to see that the doubly 2 infinite matrix (αiy) of (4) is not a Hadamard multiplier of / (Z) that we now sketch. It is enough to show that (αo ) does not induce an operator a onj he trace class matrices SΓ on /2(Z). Now & is isometric with 2 2 / (Z)(g)/ (Z) so the product gives rise to the map (ρS)k =

Σj_, «fc*ϊ of 3Γ into Λ(Z) and in fact A(Z) is isometric with J*/Ker(p). Clearly a (Ker (p)) C Ker (p) so a lifts to a' G 38 (A (Z)). Here a' is the operation of multiplication by the characteristic function of Z+ and it is THE RANGE OF A NORMAL DERIVATION 113 well known that A(Z) is not closed under this operation. (V(Ί) is not closed under the Hubert transform.) The harmonic analysis used here appears in [13; pp. 80-81] and [9; p. 64]. Hadamard multipliers are important for studying the range of a derivation mainly because of the following simple fact: (Recall that a diagonal operator is an operator for which there is an orthonormal basis of eigenvectors.)

LEMMA (3.3). Suppose A ESft(ffl) is a diagonal operator with matrix diag(λ1? A2, •) with respect to the orthonormal basis {£}. If

BE®(2e) then5fc(δB)C$ίl(δA) if and only if B = diag(/z,,μ2, •) with respect to {£} and (γή ) is an H-multiplier of 38(20 where

= (μιμi)(ι/r / ,^, Ύii 1 0 if λ, = Ay *

Proof Suppose Θl(δB)C0l(δA). Then Be {A}" by (2.5) and hence B has a diagonal matrix with respect to the given basis. If

XE 38(20 there is ZE 38(20 with δB(X) = δA(Z). Computing (/,/) entries yields (μ{ - μ})(Xξh£) = (A, - λ}){Zξh£). Now we can choose Z so that P(Z) = 0 where P is a norm one projection from 53(20 onto

{AY For such a choice Lemma (2.1) shows that (Zξh £) = 0 whenever

λi=λj and therefore {Zξh ξt) ^ yή{Xξh ξ,) with γl7 defined as in the statement of the Lemma. This proves that the multiplier condition is necessary for ί%(δβ)Cί%(δΛ) and it is clear that it is also sufficient.

LEMMA (3.4). Let μ be a finite measure on C and let A be the operator h{z)-+zh{z) on L\μ). Suppose that f E C(σ(A)) and that

£%(δ/(Λ)) CM(δA). If B is a diagonal operator with distinct eigenvalues

λ,,λ2, inσ(A) then

Proof Let {£} be an orthonormal basis of X such that JB£ = λ(£ for ί = l,2, •••. Fix an integer n>0 and let

NuN2, ,Nn of the λh each having diameter at most n~\ Finally let

U = Un be a partial isometry from ffl into L\μ) with

By (2.6) there is a map α from SΓ = J{L\μ)) into itself such that

(δ/(Λ)|^") = a(SA \3f) = 8Aay and this induces a map β from #"(30 into itself, namely β(T) = U*a(UTU*)U. An easy calculation gives 0(6 114 B. E. JOHNSON AND J. P. WILLIAMS

)-1 f ί λ(z,w)dμ(z)dμ(w) 1 ^ ij g n β\i}=[ JN> J* i>n or j>

and λ(z,w) = (f(z)-f(w))(z- w)~\l-δz,w). Now if Γn is the n) Hadamard multiplication on SΓ{ffl) associated with the matrix (β ; ) it follows that Γn and β agree on the operators ξt (g) § and since both operators are ultraweakly continuous we have ||Γn|| = \\β ||S ||α||. Hence the matrix (γjj0) defined by yf = βf- 8φf has mul- tiplier norm at most 2||α||. Since γ^-^γo where

- λ, ) ' for 17^ j Ύii 1 0 for /=/

it follows from (3.2) that (γo ) is an //-multiplier on SΓ(ffl) with multiplier norm at most 2\\a ||. Thus (δ/(β)| &) = aB(8B \ 3Γ) = δBaB for an operator aB on 2Γ(d€) with norm at most 2||α|| so that <Ά(δHB))C0l(δB).

Our next result shows that the question whether ^(δ/(A))C^(δΛ) depends only on the values of / on σ(A) and not on the normal operator A itself:

COROLLARY (3.5). Let A and f be as in the statement of the

Lemma. If B is any normal operator with σ(B)Cσ(A) then 0t(8f(B)) C

Proof. Any normal operator B on ffl is a countable direct sum of operators h{z)->zh{z) on V spaces and each of these is uniformly approximable by diagonal operators (approximate the identity function by a simple function.) Hence B itself is the uniform limit of a sequence {/?„} of diagonal operators on %C and clearly we may also choose the Bn to have distinct diagonal entries for each n. Now for each n there is an operator an on 5"($?) with (8HBn)\SΓ) = an(δBn \SΓ) =

8Bnan and supn || an \\ < oo by the proof of the Lemma. Fix XEffl(ffl) and let Zn =α*(X). Since the Zn are bounded we can pass to a subsequence if necessary to insure that the sequence {Zn} converges ultraweakly to some Z G 38 (^). Then

δfiB)(X) = (f(B) - f(Bn))X + (Bn-B)Zn + (BZn -ZnB) + Zn(B - Bn)

+ X(f(BΛ)-f(B))

so that, taking ultraweak limits, 8f{B)(X) = 8B(Z) G9t(δB) as required.

THEOREM (3.6). Let Abe a normal operator on $? with dominating scalar spectral measure μ (see [5; Theorem 10, p. 916].). IfB G THE RANGE OF A NORMAL DERIVATION 115

then 3l(δB)C0i(δA) if and only if B =f(A) where /eC(σ(A)) and (at)(z, w) = {f(z)-f{w))(z - w)~ιt(z, w) (taken as 0 when z = w) is a trace class kernel with respect to μ whenever t(z, w) is such a kernel.

Proof. Suppose first that A is the operator /z(z)—>z/i(z) on

L\μ). If 0i{δB)C0ί{δA) then Be {A}" by Theorem (2.5) so that B = f(A) for some /GL°°(μ). Also, by (2.5) there is an operator a from the trace class operators SΓ on L2(μ) into itself such that

(δB ISΓ) = a(δA 12Γ) = (δA 13T)a. If t(z,w) is the kernel of a trace class operator on L2(μ) then (z - w)(at)(z, w) = (/(z)- f(w))t(z, w) almost everywhere μ x μ. Now if μ has mass at z0 and ξ0 is the corresponding normalized characteristic function, then (at)(z0,z0) = (ξo®£o,at) = (« *(£o ® £o), 0 = 0 because the range of a * is contained in the range of 1 - P, where P is the projection of 33($?) onto {A}' used to construct a, and therefore 0 = Pa *(ξo

Consider now the general case. If 0l(δB)C0i(δA) then there is a reducing subspace % of ffl on which A is unitarily equivalent to the 2 operator h (z) —» zh (z) on L (μ). The subspace 3ίf0 reduces B and since the first art th δB(2ί?o)CδΛ(^o) P of e argument shows that B=f(A) where / has the asserted properties. Conversely if B = f(A) with / of the given form then ^(δβjC

^(δΛo) where A0,B0 are the restrictions of A,B to Xo. Corollary (3.5) then implies that $ί(δB) C&ί(δA) where A, B are the direct sum of countably many copies of Ao and Bo respectively. Since A is the restriction of A to a reducing subspace it follows that 01 (δB) C 91 (δA).

COROLLARY (3.7). Let A be a normal operator on %C and let

B E9S(X). Then 0ί{δB)C0ί(δA) if and only if there is a constant c such that \\δB(X)\\^c\\δA(X)\\ for all X<

Proof. Suppose || δβ (X) \\^c\\ δA (X) || for all X. Then BE {A}" so that B = f(A) for some bounded Borel function /. Also if T E 3~ then δB(T) - δA(S) for some S G J by (1.4) and, in the notation of (3.6), at is the kernel of the trace class operator Si satisfying 116 B. E. JOHNSON AND J. P. WILLIAMS

for all KeX, i.e., S, = S-(P*(S)|3Γ). Thus 9KδB)C9l(δA). The converse is clear from (2.6).

4. We now consider the problem of determining the elements of C{σ{A)) that satisfy the multiplier condition of Theorem (3.6). It seems that, just as there is no satisfactory way of recognizing which periodic functions have absolutely convergent Fourier series, so no easy description of the class of functions in Theorem (3.6) exists.

THEOREM (4.1). Let μ be a finite measure on C of compact support and let f G L°°(μ) satisfy the criterion of Theorem (3.6). Then f is equal a.e. μ to a continuous function which satisfies a Lipschitz condition and is differentiable relative to σ{A) at each nonisolated point ofσ(A).

Proof Let Λ(z, w) = (f(z)-f(w))(z - w)~ι for z/ w and Λ(z, z) = 0. The function Λ is μ -measurable and if ξ, η are unit vectors in L\μ) then the trace class norm of the operator with kernel Λ(z, w)ξ(z)η(w) is no less than its Hilbert-Schmidt norm

Uf\A(z,w)\2\ξ(z)\2\η(w)\2dμ(z)dμ(w)γ for which the upper bound as ξ, η vary is K = ess sup |Λ(z, w)\. Thus Λ G L\μ x μ) and | f(z) - f(w)\ ^ K \z - w | for almosf all (z, w). Put E ={z GSupp(μ): \f(z)-f(w)\ ^ K \z - w | for almost all wGSupp(μ)}.

The complement of E is of measure 0. If zί9z2EE we have

except for values of w in a set of measure 0. If μ ({zj) = 0 then we can find a sequence of values of w outside this exceptional set converging to z, so |/(z,)-/(z2)|^K|z,-z2|. If μ({z,})^0 we get the same ine- quality from the fact that z2 belongs to E. As E is dense in Supp(μ) and / is uniformly continuous on E there is a continuous function g on Supp(μ) equal to / on E, that is, equal to / a.e. By continuity \g(z)-g(w)\^K \z - w\ for z, w in σ(A). Suppose now that λ is a nonisolated point of σ{A) and that g is not differentiable at λ. Then, as g is a Lipschitz function, we can find a THE RANGE OF A NORMAL DERIVATION 117 sequence {λ, } (- oo < ϊ < oo) of distinct points in σ(A) - {λ} with A, —• λ as i -» ± oo and

\im

Replacing g by h(z) = (g(z)-pz)(q - p)~ι if necessary we can assume that p = 0, q = 1 since g satisfies the criterion of Theorem (3.6) if and only if h does. If

] = ί(g(λi)-g(λj))(λi-λj)- for iV/ γ" 1 0 for ί = j

2 then (γo ) is an Hadamard multiplier of $ (/ (Z)) by (3.3) and (3.4). This implies that the matrix (β/7) where βi7 = y-u for i,j g 1 is an //-multiplier 2 + of ^(/ (Z )). Indeed if {

lim lim ft, = 0, lim lim β/y = 1 we can find one-to-one maps π, σ from Z+ into itself such that

] \β^i)Mi)-l\/δl 10^0)1 or1 for l^i^/.

(See [8; p. 694].) Since (β^o.σo)) is clearly an //-multiplier with multiplier norm at most equal to the multiplier norm of the matrix (jf^-), and since n can be taken arbitrarily large here, it follows from (3) of

Lemma (3.2) that (α0) is a Hadamard multiplier where

_ ί 1 for i > j 10 for ί^Γ

This contradicts (4) of Lemma (3.2) and completes the proof. US B. E. JOHNSON AND J. P. WILLIAMS

COROLLARY (4.2). Suppose that f is continuous on the closed unit disk D and that £%(δ/(Λ))C£%(δΛ) for some normal operator A with σ{A) = D. Then f is analytic at each point of D. In fact, f and its derivative f belong to if00,/ satisfies the Lipschitz condition \f(eia)- f(eiβ)\ iK \a - β I on the unit circle, and the Taylor series of f is absolutely convergent on D.

Proof That / and /' belong to H00 is clear from the theorem. The other assertions about / are consequence of these, a theorem of Hardy aad Littlewood, and Hardy's inequality. See [6; pp. 48, 78]. There is a natural anti-involution τ->τ* on 55 ($(20) defined by τ*(X) = τ(X*)* for X e 38(30. With respect to this involution it is easy to check that δΛ and (δΛ)* commute if and only if A is normal so that the term "normal derivation" is unambiguous. It is known [2] that normal derivations on 3δ($?) exhibit some of the properties of normal operators on 3ίf, for example the orthogonality of range and kernel mentioned earlier. But Theorem (4.1) indicates that whereas a normal operator on $f and its adjoint always have the same range, this property fails in general for normal derivations, even those induced by diagonal operators, because z —»z is not analytic. (However, the ranges of δA and δΛ* have the same norm closure. In fact, $fc(δB)CSfc(δAy for any B in the C*-algebra generated by the normal operator A.) This fact can be expressed in a slightly different way to provide a negative answer to a question raised in a conversation with the authors of [3].

COROLLARY (4.3). There exist a diagonal operator A (with distinct eigenvalues) and a sequence {Xn} in <3l(3€) such that \\AXn - XnA ||—>0 but ||AΛΓ*-X*A||>1 for each n = l,2, .

Proof. There exists a diagonal operator A with 0ϊ(δΛ*) not contained in 2fc(δA) by the preceding remarks. For such a choice of A

Corollary (3.7) implies that for each n there is an operator Zn with

|| δΛ * (Zn) II > n || δA (Zn) ||. Then δΛ (Zn) ^ 0 by the Fuglede theorem so the choice Xn = Zn/n \\δΛ(Zn)\\ satisfies the required conditions.

REMARK. The sequence {Xn} cannot be chosen to be uniformly bounded however [R. L. Moore, private communication.] In [3] a counterexample is constructed to show that if A is a normal operator on $f and P is a projection in {A}", then in general one cannot find a positive number δ corresponding to each e > 0 so that the conditions X E S& (%), \\ AX - XA || < δ imply || PX - XP \\ < e. Or e- quivalently, by (3.7), the condition P G {A}" is not sufficient for 3$(δP) C

έ%(δΛ). Theorem (4.1) helps to explain this situation more fully: THE RANGE OF A NORMAL DERIVATION 119

COROLLARY (4.4). Let A be a normal operator on $f with spectral measure E( ) and let P be a projection on X. Then &(8P)C0l(8A) if and only if there are disjoint closed sets Δo, Δi with Δo U Δi = σ(A) and P = £(Δ,).

Proof If 9i{8P)CSk{8A) then by (4.1) there is a continuous function / on σ(A) with P = f(A). The spectral mapping * theorem implies that σ{A) is the union of the disjoint closed sets Δo = /"'(0) and ! Δ, = /- (1). Also P = f(A) = Jf(λ)dEk = E(Δ,).

Conversely if P = f?(Δi) where Δo, Δj are disjoint closed sets whose union is the spectrum of A then by the Riesz-Dunford P - f(A) for some function / that is analytic in a neighborhood of σ(A). Hence 94(δP) = &(δ/(A))C94(δΛ) by (3.6) or by the result of Weber [20] mentioned in the Introduction. The sufficiency of the condition on P may also be established directly by considering the decomposition 3€ = 2?(Δ0)3ίf 0f?(Δ,)$ί and appealing to the theorem of Lumer and Rosenblum [11] on the solvabil- ity of the operator equations AιZ-ZA0 = X9 A0Y-YAX= W for operators AQ,A\ with disjoint spectra. A theorem of Anderson [1] shows that if N is a normal operator then 0l(8N)C U 2fc(8A) where the union is taken over the set of all self adjoint operators A in £$($?). That is, any commutator of the form NX-XN can also be written AY-YA for some A = A* and YεS8(Sif). Theorem (4.1) implies that one cannot improve this to: gi{8N)C<3l{8A) for some A = A*. Indeed if σ(N) has infinite one- dimensional Hausdorff measure then we cannot have σ(N) = /(σ(A)), and hence cannot have N - /(A), for any self-adjoint A and Lipschitz /. Theorem (4.1) also permits a somewhat simpler proof of the theorem of Anderson and Foias [1] which determines when the range of a normal derivation is closed.

COROLLARY (4.5). Let Abe a normal operator on $f. Then 9h (δΛ) is norm closed in 8ft (2£) if and only if the spectrum of A is finite.

Proof Suppose that 34 (δΛ) is norm closed in 9i(3€). If P is a norm 1 projection of 35 ($ί) onto the commutant of A then there is a constant c >0 such that ||δΛ(X)||^ c \\X\\ for all Xe»(l-P). For B E {A}" and X G &(%) we have P(BX -XB) = BP(X) - P(X)B = 0 and so 94(δB)C94(l-P). Hence

c || δB (X) || ^ || δΛ (δB (X)) II = II δB (δA (X)) II έ II δB IIII8A (X) II. 120 B. E. JOHNSON AND J. P. WILLIAMS

Therefore &ί(δB)Cίk{δA) by (3.7) so that B = f(Λ) for some continuous function / by (3.6). Thus {A}'\ the generated by A, coincides with the C*-algebra generated by A. It follows that σ(A) is extremally disconnected and therefore is a finite subset of C. Conversely if σ(A) is finite then A is a linear combination of

orthogonal projections and this easily implies that δΛ has closed range.

REMARK. With a different argument one can show that if A is r normal with infinite spectrum then 2ft(δA) + {A} is not norm dense in

38(30. (See [2].) Hence 0t(δA)£9t(l -P) for any projection P of norm 1 from 38(30 onto {A}'. We conclude this section with an example that confirms a remark of J. Taylor [18; p. 29].

EXAMPLE (4.6). Let A be the operator of multiplication by the I 2 sequence λ, = /" (iV0), λ0 = 0 in / (Z), and let / be the Lipschitz + function f(χ) = χ = \(χ + |JC|). Then 9l{δίiA))(tS/i{δA). That is, there does not exist a constant c such that \\f(A)X - Xf(A)\\^c\\AX - XA || for all operators X on /2(Z). This follows at once from (4.1) since / is not differentiate at x = 0. The result can also be proved directly by observing that

1 „ =ί(/(I)/(i))(ίλ/)- if IVJ Ύii lO if i=j

cannot be an //-multiplier of 35(/2(Z)) because this would imply that the matrix (JO'+jT1) is an //-multiplier of 39(/2(Z+)) which is impossible because of the relations

lim /(/+/)"• = 1, lim j(ί + j)"1 = 0.

(See the proof of (4.1).)

5. We conclude by giving a condition which is sufficient for / to satisfy the criterion of Theorem (3.6) for self-adjoint operators.

THEOREM (5.1). Let f be a complex valued function on R with continuous third derivative. If A is a self-adjoint operator on X then

Proof. By (3.5) it is sufficient to prove the theorem for the case in which A is the operator f(x)^>x - f(x) on L\I) where / is a compact subinterval of R. Define THE RANGE OF A NORMAL DERIVATION 121

x = y.

Then Λ is a C(2) function on R x R and, without altering Λ in a neighborhood of σ{A) x σ(Λ) we can assume that Λ is doubly periodic

ι ι with periods p,q say. The Fourier coefficients λkl =p~ q~ \ Jo Jo 1 ι 2 2 2 2 2 Λ(JC, y)exp2ττi(/cjcp" + lyq)dxdy satisfy {k λkl} E / (Z ), {/%} G / (Z ) because d2A/dx2 and <92Λ/<9y2 belong to V. Since {(fc2 + Z2)'1}E /2(Z2) it 2 follows that {λkl}El\Z ). Now if t (x, y) is the kernel of a trace class operator T on L2(/) then ι (akιt)(x,y) = t(x,y)exp(-2τri(kxp~ + lyq"*)) is the kernel of the

operator UTV where U and V are unitary so au is an operator of norm

1 in ί$(ίf). Hence a = Σwλwαw is an operator in £$(50 and this is ι 1 t —>Λf because A = Σλw exp(- 2πi(kxp~ + /yq" )). Thus At is a trace class kernel whenever ί is and it follows as in the proof of Theorem (3.6) that 9t(δfiA))C9t(δA).

Note that now that we know <3i(δnA)) C$t(δA) Theorem (3.6) shows that / also satisfies the multiplier condition of that theorem. That is, the function Λ(z, w), taken to be 0 when z — w, rather than /'(z), also multiplies trace class kernels.

REFERENCES

1. J. H. Anderson, Derivations, commutators, and the essential , Ph.D. thesis, Indiana University, 1971. 2. , On normal derivations, Proc. Amer. Math. Soc, 38 (1973), 135-140. 3. J. J. Bastian, K. J. Harrison, Subnormal weighted shifts and asymptotic properties of normal operators, Proc. Amer. Math. Soc, 42 (1974), 475-479. 4. J. Dixmier, Les fonctionelles lineaires sur Γensemble des operateurs bornesmd'un espace de Hilbert, Ann. Math., 51 (1950), 387-408. 5. N. Dunford, J. T. Schwartz, Linear Operators, Part II, Interscience, New York, 1963. 6. P. L. Duren, Theory of Hp Spaces, Academic Press, New York, 1970. 7. B. E. Johnson, Continuity of linear operators commuting with continuous linear operators, Trans. Amer. Math. Soc, 128 (1967), 88-102. 8. , Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math., 94 (1972), 685-698. 9. Y. Katznelson, An introduction to Harmonic Analysis, Wiley, New York, 1968. 10. V. B. Korotkov, Integral operators with Carleman kernels, Doklady Akad. Nauk SSSR 165 (1965), 1496-1499. 11. G. Lumer, M. Rosenblum, Linear operator equations, Proc Amer. Math. Soc, 10 (1959), 32-41. 12. J. von Neumann, On rings of operators HI, Ann. of Math. 41 (1940), 94-161. 13. M. A. Rieffel, Multipliers and tensor products of Lp spaces of locally compact groups, Studia Math. 33 (1969), 71-82. 14. R. Schatten, A theory of cross-spaces, Ann. Math. Studies No. 26, Princeton, 1950. 15. I. Schur, Bemerkungen zur Theorie der beschrάnkten Bilinearformen mit unendlich vielen Verάnderlichen, J. Reine Angew. Math., 140 (1911), 1-28. 16. J. G. Stampfli, Derivations on &(%)'. The range, Illinois J. Math., 17 (1973), 518-524. 122 B. E. JOHNSON AND J. P. WILLIAMS

17. J. Tomiyama, On the projection of norm one in W*-algbras, Proc. Japan Acad., 33 (1957), 608-612. 18. J. L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc, 79 (1973), 1-34. 19. R. E. Weber, Derivation ranges, Ph.D. thesis, Indiana University, 1972. 20. , Analytic functions, ideals, and derivation ranges, Proc. Amer. Math. Soc. 21. J. Weidmann, Carlemanoperatoren, Manuscripta Math. 2 (1970), 1-38. 22. J. P. Williams, On the range of a derivation, Pacific J. Math., 38 (1971), 273-279. 23. , On the range of a derivation II, to appear in Proc. Roy. Irish Acad.

Received November 19, 1973 and in revised form May 21, 1974. The second author would like to express his gratitutde to the Science Research Council and to the University of Newcastle upon Tyne for their financial support and hospitality during the period of this research.

THE UNIVERSITY, NEWCASTLE UPON TYNE, ENGLAND AND INDIANA UNIVERSITY CONTENTS

Zvi Artstein and John A. Burns, Integration of compact set-valued functions 297 J. A. Beachy and W. D. Blair, Rings whose faithful left ideals are cofaithful 1 Mark Benard, Characters and Schur indices of the unitary reflection group [321]3 309 H. L. Bentley and B. J. Taylor, Wallman rings 15 E. Berman, Matrix rings over polynomial identity rings II 37 Simeon M. Berman, A new characterization of characteristic functions of absolutely continuous distributions 323 Monte B. Boisen, Jr. and Philip B. Sheldon, Pre-Prufer rings 331 A. K. Boyle and K. R. Goodearl, Rings over which certain modules are injective 43 J. L. Brenner, R. M. Crabwell and J. Riddell, Covering theorems for finite nonabelian simple groups. V 55 H. H. Brungs, Three questions on duo rings 345 Iracema M. Bund, Birnbaum-Orlicz spaces of functions on groups ....351 John D. Elwin and Donald R. Short, Branched immersions between 2-manifolds ofhigher topological type 361 J. K. Finch, The single valued extension property on a Banach space 61 J. R. Fisher, A Goldie theorem for differentiably prime rings 71 Eric M. Friedlander, Extension functions for rank 2, torsion free abelian groups 371 J. Froemke and R. Quackenbusch, The spectrum of an equational class of groupoids 381 B. J. Gardner, Radicals of supplementary semilattice sums of associative rings 387 Shmuel Glasner, Relatively invariant measures 393 G. R. Gordh, Jr. and Sibe Mardesic, Characterizing local connected- ness in inverse limits 411 S. Graf, On the existence of strong liftings in second countable topological spaces 419 S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces 427 F. Hansen, On one-sided prime ideals 79 D. J. Hartfiel and C. J. Maxson, A characterization of the maximal monoids and maximal groups in βx 437 Robert E. Hartwig and S. Brent Morris, The universal flip matrix and the generalized faro-shuffle 445 aicJunlo ahmtc 95Vl 8 o 1 No. 58, Vol. 1975 Mathematics of Journal Pacific Pacific Journal of Mathematics Vol. 58, No. 1 March, 1975

John Allen Beachy and William David Blair, Rings whose faithful left ideals are cofaithful ...... 1 Herschel Lamar Bentley and Barbara June Taylor, Wallman rings ...... 15 Elizabeth Berman, Matrix rings over polynomial identity rings. II ...... 37 Ann K. Boyle and Kenneth R. Goodearl, Rings over which certain modules are injective ...... 43 J. L. Brenner, Robert Myrl Cranwell and James Riddell, Covering theorems for finite nonabelian simple groups. V ...... 55 James Kenneth Finch, The single valued extension property on a Banach space ...... 61 John Robert Fisher, A Goldie theorem for differentiably prime rings...... 71 Friedhelm Hansen, On one-sided prime ideals ...... 79 Jon Craig Helton, Product integrals and the solution of integral equations ...... 87 Barry E. Johnson and James Patrick Williams, The range of a normal derivation ...... 105 Kurt Kreith, A dynamical criterion for conjugate points...... 123 Robert Allen McCoy, Baire spaces and hyperspaces ...... 133 John McDonald, Isometries of the ...... 143 H. Minc, Doubly stochastic matrices with minimal permanents ...... 155 Shahbaz Noorvash, Covering the vertices of a graph by vertex-disjoint paths...... 159 Theodore Windle Palmer, Jordan ∗-homomorphisms between reduced Banach ∗-algebras ...... 169 Donald Steven Passman, On the semisimplicity of group rings of some locally finite groups ...... 179 Mario Petrich, Varieties of orthodox bands of groups ...... 209 Robert Horace Redfield, The generalized interval topology on distributive lattices ...... 219 James Wilson Stepp, Algebraic maximal semilattices ...... 243 Patrick Noble Stewart, A sheaf theoretic representation of rings with Boolean orthogonalities ...... 249 Ting-On To and Kai Wing Yip, A generalized Jensen’s inequality ...... 255 Arnold Lewis Villone, Second order differential operators with self-adjoint extensions ...... 261 Martin E. Walter, On the structure of the Fourier-Stieltjes algebra ...... 267 John Wermer, Subharmonicity and hulls ...... 283 Edythe Parker Woodruff, A map of E3 onto E3 taking no disk onto a disk ...... 291